\(\int (a+b \csc (e+f x))^m \sin (e+f x) \, dx\) [58]

Optimal result

Integrand size = 19, antiderivative size = 19 \[ \int (a+b \csc (e+f x))^m \sin (e+f x) \, dx=\text {Int}\left ((a+b \csc (e+f x))^m \sin (e+f x),x\right ) \]

[Out]

Unintegrable((a+b*csc(f*x+e))^m*sin(f*x+e),x)

Rubi [N/A]

Not integrable

Time = 0.04 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int (a+b \csc (e+f x))^m \sin (e+f x) \, dx=\int (a+b \csc (e+f x))^m \sin (e+f x) \, dx \]

[In]

Int[(a + b*Csc[e + f*x])^m*Sin[e + f*x],x]

[Out]

Defer[Int][(a + b*Csc[e + f*x])^m*Sin[e + f*x], x]

Rubi steps \begin{align*} \text {integral}& = \int (a+b \csc (e+f x))^m \sin (e+f x) \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 8.30 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.11 \[ \int (a+b \csc (e+f x))^m \sin (e+f x) \, dx=\int (a+b \csc (e+f x))^m \sin (e+f x) \, dx \]

[In]

Integrate[(a + b*Csc[e + f*x])^m*Sin[e + f*x],x]

[Out]

Integrate[(a + b*Csc[e + f*x])^m*Sin[e + f*x], x]

Maple [N/A] (verified)

Not integrable

Time = 0.68 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00

\[\int \left (a +b \csc \left (f x +e \right )\right )^{m} \sin \left (f x +e \right )d x\]

[In]

int((a+b*csc(f*x+e))^m*sin(f*x+e),x)

[Out]

int((a+b*csc(f*x+e))^m*sin(f*x+e),x)

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.11 \[ \int (a+b \csc (e+f x))^m \sin (e+f x) \, dx=\int { {\left (b \csc \left (f x + e\right ) + a\right )}^{m} \sin \left (f x + e\right ) \,d x } \]

[In]

integrate((a+b*csc(f*x+e))^m*sin(f*x+e),x, algorithm="fricas")

[Out]

integral((b*csc(f*x + e) + a)^m*sin(f*x + e), x)

Sympy [N/A]

Not integrable

Time = 4.48 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int (a+b \csc (e+f x))^m \sin (e+f x) \, dx=\int \left (a + b \csc {\left (e + f x \right )}\right )^{m} \sin {\left (e + f x \right )}\, dx \]

[In]

integrate((a+b*csc(f*x+e))**m*sin(f*x+e),x)

[Out]

Integral((a + b*csc(e + f*x))**m*sin(e + f*x), x)

Maxima [N/A]

Not integrable

Time = 1.65 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.11 \[ \int (a+b \csc (e+f x))^m \sin (e+f x) \, dx=\int { {\left (b \csc \left (f x + e\right ) + a\right )}^{m} \sin \left (f x + e\right ) \,d x } \]

[In]

integrate((a+b*csc(f*x+e))^m*sin(f*x+e),x, algorithm="maxima")

[Out]

integrate((b*csc(f*x + e) + a)^m*sin(f*x + e), x)

Giac [N/A]

Not integrable

Time = 0.40 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.11 \[ \int (a+b \csc (e+f x))^m \sin (e+f x) \, dx=\int { {\left (b \csc \left (f x + e\right ) + a\right )}^{m} \sin \left (f x + e\right ) \,d x } \]

[In]

integrate((a+b*csc(f*x+e))^m*sin(f*x+e),x, algorithm="giac")

[Out]

integrate((b*csc(f*x + e) + a)^m*sin(f*x + e), x)

Mupad [N/A]

Not integrable

Time = 18.43 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.21 \[ \int (a+b \csc (e+f x))^m \sin (e+f x) \, dx=\int \sin \left (e+f\,x\right )\,{\left (a+\frac {b}{\sin \left (e+f\,x\right )}\right )}^m \,d x \]

[In]

int(sin(e + f*x)*(a + b/sin(e + f*x))^m,x)

[Out]

int(sin(e + f*x)*(a + b/sin(e + f*x))^m, x)